Kitaev's quantum double model from a local quantum physics point of view

TL;DR

This paper examines Kitaev's quantum double models from a local quantum physics perspective, analyzing superselection sectors, anyons, and potential applications in quantum computation.

Contribution

It provides a local quantum physics analysis of Kitaev's quantum double models, including superselection sectors, anyon statistics, and extensions to non-abelian groups.

Findings

Superselection sectors correspond to representations of the quantum double

Anyons exhibit non-trivial braiding phases

Potential applications in topological quantum computation

Abstract

A prominent example of a topologically ordered system is Kitaev's quantum double model $\mathcal{D}(G)$ for finite groups $G$ (which in particular includes $G = \mathbb{Z}_2$, the toric code). We will look at these models from the point of view of local quantum physics. In particular, we will review how in the abelian case, one can do a Doplicher-Haag-Roberts analysis to study the different superselection sectors of the model. In this way one finds that the charges are in one-to-one correspondence with the representations of $\mathcal{D}(G)$, and that they are in fact anyons. Interchanging two of such anyons gives a non-trivial phase, not just a possible sign change. The case of non-abelian groups $G$ is more complicated. We outline how one could use amplimorphisms, that is, morphisms $A \to M_n(A)$ to study the superselection structure in that case. Finally, we give a brief overview of applications of topologically ordered systems to the field of quantum computation.